Graphics.Rasterific.Shading:$sgradientColorAt from Rasterific-0.6.1

Percentage Accurate: 100.0% → 100.0%

Time: 6.9s

Alternatives: 9

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x - y}{z - y} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (- x y) (- z y)))

C

double code(double x, double y, double z) {
	return (x - y) / (z - y);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x - y) / (z - y)
end function

Java

public static double code(double x, double y, double z) {
	return (x - y) / (z - y);
}

Python

def code(x, y, z):
	return (x - y) / (z - y)

Julia

function code(x, y, z)
	return Float64(Float64(x - y) / Float64(z - y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x - y) / (z - y);
end

Wolfram

code[x_, y_, z_] := N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - y}{z - y} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (- x y) (- z y)))

C

double code(double x, double y, double z) {
	return (x - y) / (z - y);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x - y) / (z - y)
end function

Java

public static double code(double x, double y, double z) {
	return (x - y) / (z - y);
}

Python

def code(x, y, z):
	return (x - y) / (z - y)

Julia

function code(x, y, z)
	return Float64(Float64(x - y) / Float64(z - y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x - y) / (z - y);
end

Wolfram

code[x_, y_, z_] := N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - y}{z - y} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (- x y) (- z y)))

C

double code(double x, double y, double z) {
	return (x - y) / (z - y);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x - y) / (z - y)
end function

Java

public static double code(double x, double y, double z) {
	return (x - y) / (z - y);
}

Python

def code(x, y, z):
	return (x - y) / (z - y)

Julia

function code(x, y, z)
	return Float64(Float64(x - y) / Float64(z - y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x - y) / (z - y);
end

Wolfram

code[x_, y_, z_] := N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{x - y}{z - y} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 97.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z - y}\\ t_1 := \frac{x}{z - y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- z y))) (t_1 (/ x (- z y))))
   (if (<= t_0 -1e+23)
     t_1
     (if (<= t_0 2e-15) (/ (- x y) z) (if (<= t_0 2.0) (/ y (- y z)) t_1)))))

C

double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double t_1 = x / (z - y);
	double tmp;
	if (t_0 <= -1e+23) {
		tmp = t_1;
	} else if (t_0 <= 2e-15) {
		tmp = (x - y) / z;
	} else if (t_0 <= 2.0) {
		tmp = y / (y - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x - y) / (z - y)
    t_1 = x / (z - y)
    if (t_0 <= (-1d+23)) then
        tmp = t_1
    else if (t_0 <= 2d-15) then
        tmp = (x - y) / z
    else if (t_0 <= 2.0d0) then
        tmp = y / (y - z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double t_1 = x / (z - y);
	double tmp;
	if (t_0 <= -1e+23) {
		tmp = t_1;
	} else if (t_0 <= 2e-15) {
		tmp = (x - y) / z;
	} else if (t_0 <= 2.0) {
		tmp = y / (y - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x - y) / (z - y)
	t_1 = x / (z - y)
	tmp = 0
	if t_0 <= -1e+23:
		tmp = t_1
	elif t_0 <= 2e-15:
		tmp = (x - y) / z
	elif t_0 <= 2.0:
		tmp = y / (y - z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - y) / Float64(z - y))
	t_1 = Float64(x / Float64(z - y))
	tmp = 0.0
	if (t_0 <= -1e+23)
		tmp = t_1;
	elseif (t_0 <= 2e-15)
		tmp = Float64(Float64(x - y) / z);
	elseif (t_0 <= 2.0)
		tmp = Float64(y / Float64(y - z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x - y) / (z - y);
	t_1 = x / (z - y);
	tmp = 0.0;
	if (t_0 <= -1e+23)
		tmp = t_1;
	elseif (t_0 <= 2e-15)
		tmp = (x - y) / z;
	elseif (t_0 <= 2.0)
		tmp = y / (y - z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+23], t$95$1, If[LessEqual[t$95$0, 2e-15], N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -9.9999999999999992e22 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x}{z - y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{z - y}} \]

      2. lower--.f64 99.9

         \[\leadsto \frac{x}{\color{blue}{z - y}} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{x}{z - y}} \]

   if -9.9999999999999992e22 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2.0000000000000002e-15

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{x - y}{z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x - y}{z}} \]

      2. lower--.f64 99.9

         \[\leadsto \frac{\color{blue}{x - y}}{z} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{x - y}{z}} \]

   if 2.0000000000000002e-15 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{y}{z - y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{y}{z - y}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{y}{\mathsf{neg}\left(\left(z - y\right)\right)}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\mathsf{neg}\left(\left(z - y\right)\right)}} \]

      4. sub-neg N/A

         \[\leadsto \frac{y}{\mathsf{neg}\left(\color{blue}{\left(z + \left(\mathsf{neg}\left(y\right)\right)\right)}\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{y}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + z\right)}\right)} \]

      6. distribute-neg-in N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + \left(\mathsf{neg}\left(z\right)\right)}} \]

      7. remove-double-neg N/A

         \[\leadsto \frac{y}{\color{blue}{y} + \left(\mathsf{neg}\left(z\right)\right)} \]

      8. *-rgt-identity N/A

         \[\leadsto \frac{y}{y + \left(\mathsf{neg}\left(\color{blue}{z \cdot 1}\right)\right)} \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \frac{y}{y + \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot 1}} \]

      10. *-inverses N/A

          \[\leadsto \frac{y}{y + \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\frac{y}{y}}} \]

      11. associate-/l* N/A

          \[\leadsto \frac{y}{y + \color{blue}{\frac{\left(\mathsf{neg}\left(z\right)\right) \cdot y}{y}}} \]

      12. distribute-lft-neg-out N/A

          \[\leadsto \frac{y}{y + \frac{\color{blue}{\mathsf{neg}\left(z \cdot y\right)}}{y}} \]

      13. distribute-rgt-neg-out N/A

          \[\leadsto \frac{y}{y + \frac{\color{blue}{z \cdot \left(\mathsf{neg}\left(y\right)\right)}}{y}} \]

      14. associate-*l/ N/A

          \[\leadsto \frac{y}{y + \color{blue}{\frac{z}{y} \cdot \left(\mathsf{neg}\left(y\right)\right)}} \]

      15. distribute-rgt-neg-in N/A

          \[\leadsto \frac{y}{y + \color{blue}{\left(\mathsf{neg}\left(\frac{z}{y} \cdot y\right)\right)}} \]

      16. unsub-neg N/A

          \[\leadsto \frac{y}{\color{blue}{y - \frac{z}{y} \cdot y}} \]

      17. remove-double-neg N/A

          \[\leadsto \frac{y}{y - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{z}{y} \cdot y\right)\right)\right)\right)}} \]

      18. distribute-lft-neg-out N/A

          \[\leadsto \frac{y}{y - \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{z}{y}\right)\right) \cdot y}\right)\right)} \]

      19. mul-1-neg N/A

          \[\leadsto \frac{y}{y - \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{z}{y}\right)} \cdot y\right)\right)} \]

      20. associate-*r/ N/A

          \[\leadsto \frac{y}{y - \left(\mathsf{neg}\left(\color{blue}{\frac{-1 \cdot z}{y}} \cdot y\right)\right)} \]

      21. mul-1-neg N/A

          \[\leadsto \frac{y}{y - \left(\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(z\right)}}{y} \cdot y\right)\right)} \]

      22. associate-*l/ N/A

          \[\leadsto \frac{y}{y - \left(\mathsf{neg}\left(\color{blue}{\frac{\left(\mathsf{neg}\left(z\right)\right) \cdot y}{y}}\right)\right)} \]

      23. associate-/l* N/A

          \[\leadsto \frac{y}{y - \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \frac{y}{y}}\right)\right)} \]

      24. *-inverses N/A

          \[\leadsto \frac{y}{y - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{1}\right)\right)} \]

      25. distribute-lft-neg-out N/A

          \[\leadsto \frac{y}{y - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) \cdot 1}} \]

      26. remove-double-neg N/A

          \[\leadsto \frac{y}{y - \color{blue}{z} \cdot 1} \]

      27. *-rgt-identity N/A

          \[\leadsto \frac{y}{y - \color{blue}{z}} \]

   5. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\frac{y}{y - z}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 84.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-47} \lor \neg \left(t\_0 \leq 2\right):\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- z y))))
   (if (or (<= t_0 2e-47) (not (<= t_0 2.0))) (/ x (- z y)) (/ y (- y z)))))

C

double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double tmp;
	if ((t_0 <= 2e-47) || !(t_0 <= 2.0)) {
		tmp = x / (z - y);
	} else {
		tmp = y / (y - z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - y) / (z - y)
    if ((t_0 <= 2d-47) .or. (.not. (t_0 <= 2.0d0))) then
        tmp = x / (z - y)
    else
        tmp = y / (y - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double tmp;
	if ((t_0 <= 2e-47) || !(t_0 <= 2.0)) {
		tmp = x / (z - y);
	} else {
		tmp = y / (y - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x - y) / (z - y)
	tmp = 0
	if (t_0 <= 2e-47) or not (t_0 <= 2.0):
		tmp = x / (z - y)
	else:
		tmp = y / (y - z)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if ((t_0 <= 2e-47) || !(t_0 <= 2.0))
		tmp = Float64(x / Float64(z - y));
	else
		tmp = Float64(y / Float64(y - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x - y) / (z - y);
	tmp = 0.0;
	if ((t_0 <= 2e-47) || ~((t_0 <= 2.0)))
		tmp = x / (z - y);
	else
		tmp = y / (y - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 2e-47], N[Not[LessEqual[t$95$0, 2.0]], $MachinePrecision]], N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision], N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 1.9999999999999999e-47 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x}{z - y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{z - y}} \]

      2. lower--.f64 82.3

         \[\leadsto \frac{x}{\color{blue}{z - y}} \]

   5. Applied rewrites 82.3%

      \[\leadsto \color{blue}{\frac{x}{z - y}} \]

   if 1.9999999999999999e-47 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{y}{z - y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{y}{z - y}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{y}{\mathsf{neg}\left(\left(z - y\right)\right)}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\mathsf{neg}\left(\left(z - y\right)\right)}} \]

      4. sub-neg N/A

         \[\leadsto \frac{y}{\mathsf{neg}\left(\color{blue}{\left(z + \left(\mathsf{neg}\left(y\right)\right)\right)}\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{y}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + z\right)}\right)} \]

      6. distribute-neg-in N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + \left(\mathsf{neg}\left(z\right)\right)}} \]

      7. remove-double-neg N/A

         \[\leadsto \frac{y}{\color{blue}{y} + \left(\mathsf{neg}\left(z\right)\right)} \]

      8. *-rgt-identity N/A

         \[\leadsto \frac{y}{y + \left(\mathsf{neg}\left(\color{blue}{z \cdot 1}\right)\right)} \]

      9. distribute-lft-neg-in N/A

         \[\leadsto \frac{y}{y + \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot 1}} \]

      10. *-inverses N/A

          \[\leadsto \frac{y}{y + \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\frac{y}{y}}} \]

      11. associate-/l* N/A

          \[\leadsto \frac{y}{y + \color{blue}{\frac{\left(\mathsf{neg}\left(z\right)\right) \cdot y}{y}}} \]

      12. distribute-lft-neg-out N/A

          \[\leadsto \frac{y}{y + \frac{\color{blue}{\mathsf{neg}\left(z \cdot y\right)}}{y}} \]

      13. distribute-rgt-neg-out N/A

          \[\leadsto \frac{y}{y + \frac{\color{blue}{z \cdot \left(\mathsf{neg}\left(y\right)\right)}}{y}} \]

      14. associate-*l/ N/A

          \[\leadsto \frac{y}{y + \color{blue}{\frac{z}{y} \cdot \left(\mathsf{neg}\left(y\right)\right)}} \]

      15. distribute-rgt-neg-in N/A

          \[\leadsto \frac{y}{y + \color{blue}{\left(\mathsf{neg}\left(\frac{z}{y} \cdot y\right)\right)}} \]

      16. unsub-neg N/A

          \[\leadsto \frac{y}{\color{blue}{y - \frac{z}{y} \cdot y}} \]

      17. remove-double-neg N/A

          \[\leadsto \frac{y}{y - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{z}{y} \cdot y\right)\right)\right)\right)}} \]

      18. distribute-lft-neg-out N/A

          \[\leadsto \frac{y}{y - \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{z}{y}\right)\right) \cdot y}\right)\right)} \]

      19. mul-1-neg N/A

          \[\leadsto \frac{y}{y - \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{z}{y}\right)} \cdot y\right)\right)} \]

      20. associate-*r/ N/A

          \[\leadsto \frac{y}{y - \left(\mathsf{neg}\left(\color{blue}{\frac{-1 \cdot z}{y}} \cdot y\right)\right)} \]

      21. mul-1-neg N/A

          \[\leadsto \frac{y}{y - \left(\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left(z\right)}}{y} \cdot y\right)\right)} \]

      22. associate-*l/ N/A

          \[\leadsto \frac{y}{y - \left(\mathsf{neg}\left(\color{blue}{\frac{\left(\mathsf{neg}\left(z\right)\right) \cdot y}{y}}\right)\right)} \]

      23. associate-/l* N/A

          \[\leadsto \frac{y}{y - \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \frac{y}{y}}\right)\right)} \]

      24. *-inverses N/A

          \[\leadsto \frac{y}{y - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{1}\right)\right)} \]

      25. distribute-lft-neg-out N/A

          \[\leadsto \frac{y}{y - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) \cdot 1}} \]

      26. remove-double-neg N/A

          \[\leadsto \frac{y}{y - \color{blue}{z} \cdot 1} \]

      27. *-rgt-identity N/A

          \[\leadsto \frac{y}{y - \color{blue}{z}} \]

   5. Applied rewrites 95.4%

      \[\leadsto \color{blue}{\frac{y}{y - z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 2 \cdot 10^{-47} \lor \neg \left(\frac{x - y}{z - y} \leq 2\right):\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-15} \lor \neg \left(t\_0 \leq 5 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- z y))))
   (if (or (<= t_0 2e-15) (not (<= t_0 5e+15)))
     (/ x (- z y))
     (- 1.0 (/ x y)))))

C

double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double tmp;
	if ((t_0 <= 2e-15) || !(t_0 <= 5e+15)) {
		tmp = x / (z - y);
	} else {
		tmp = 1.0 - (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - y) / (z - y)
    if ((t_0 <= 2d-15) .or. (.not. (t_0 <= 5d+15))) then
        tmp = x / (z - y)
    else
        tmp = 1.0d0 - (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double tmp;
	if ((t_0 <= 2e-15) || !(t_0 <= 5e+15)) {
		tmp = x / (z - y);
	} else {
		tmp = 1.0 - (x / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x - y) / (z - y)
	tmp = 0
	if (t_0 <= 2e-15) or not (t_0 <= 5e+15):
		tmp = x / (z - y)
	else:
		tmp = 1.0 - (x / y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if ((t_0 <= 2e-15) || !(t_0 <= 5e+15))
		tmp = Float64(x / Float64(z - y));
	else
		tmp = Float64(1.0 - Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x - y) / (z - y);
	tmp = 0.0;
	if ((t_0 <= 2e-15) || ~((t_0 <= 5e+15)))
		tmp = x / (z - y);
	else
		tmp = 1.0 - (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 2e-15], N[Not[LessEqual[t$95$0, 5e+15]], $MachinePrecision]], N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 2.0000000000000002e-15 or 5e15 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x}{z - y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{z - y}} \]

      2. lower--.f64 79.8

         \[\leadsto \frac{x}{\color{blue}{z - y}} \]

   5. Applied rewrites 79.8%

      \[\leadsto \color{blue}{\frac{x}{z - y}} \]

   if 2.0000000000000002e-15 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5e15

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x - y}{y}} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{x}{y} - \frac{y}{y}\right)} \]

      2. sub-neg N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{x}{y} + \left(\mathsf{neg}\left(\frac{y}{y}\right)\right)\right)} \]

      3. *-inverses N/A

         \[\leadsto -1 \cdot \left(\frac{x}{y} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto -1 \cdot \left(\frac{x}{y} + \color{blue}{-1}\right) \]

      5. +-commutative N/A

         \[\leadsto -1 \cdot \color{blue}{\left(-1 + \frac{x}{y}\right)} \]

      6. distribute-lft-in N/A

         \[\leadsto \color{blue}{-1 \cdot -1 + -1 \cdot \frac{x}{y}} \]

      7. metadata-eval N/A

         \[\leadsto \color{blue}{1} + -1 \cdot \frac{x}{y} \]

      8. mul-1-neg N/A

         \[\leadsto 1 + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)} \]

      9. unsub-neg N/A

         \[\leadsto \color{blue}{1 - \frac{x}{y}} \]

      10. lower--.f64 N/A

          \[\leadsto \color{blue}{1 - \frac{x}{y}} \]

      11. lower-/.f64 96.0

          \[\leadsto 1 - \color{blue}{\frac{x}{y}} \]

   5. Applied rewrites 96.0%

      \[\leadsto \color{blue}{1 - \frac{x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 2 \cdot 10^{-15} \lor \neg \left(\frac{x - y}{z - y} \leq 5 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_0 \leq 0.2:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{z}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- z y))))
   (if (<= t_0 0.2) (/ x z) (if (<= t_0 2.0) (+ (/ z y) 1.0) (/ (- x) y)))))

C

double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double tmp;
	if (t_0 <= 0.2) {
		tmp = x / z;
	} else if (t_0 <= 2.0) {
		tmp = (z / y) + 1.0;
	} else {
		tmp = -x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - y) / (z - y)
    if (t_0 <= 0.2d0) then
        tmp = x / z
    else if (t_0 <= 2.0d0) then
        tmp = (z / y) + 1.0d0
    else
        tmp = -x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double tmp;
	if (t_0 <= 0.2) {
		tmp = x / z;
	} else if (t_0 <= 2.0) {
		tmp = (z / y) + 1.0;
	} else {
		tmp = -x / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x - y) / (z - y)
	tmp = 0
	if t_0 <= 0.2:
		tmp = x / z
	elif t_0 <= 2.0:
		tmp = (z / y) + 1.0
	else:
		tmp = -x / y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_0 <= 0.2)
		tmp = Float64(x / z);
	elseif (t_0 <= 2.0)
		tmp = Float64(Float64(z / y) + 1.0);
	else
		tmp = Float64(Float64(-x) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_0 <= 0.2)
		tmp = x / z;
	elseif (t_0 <= 2.0)
		tmp = (z / y) + 1.0;
	else
		tmp = -x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.2], N[(x / z), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(z / y), $MachinePrecision] + 1.0), $MachinePrecision], N[((-x) / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 0.20000000000000001

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 61.7

         \[\leadsto \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 61.7%

      \[\leadsto \color{blue}{\frac{x}{z}} \]

   if 0.20000000000000001 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(1 + -1 \cdot \frac{x}{y}\right) - -1 \cdot \frac{z}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)}\right) - -1 \cdot \frac{z}{y} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\left(1 - \frac{x}{y}\right)} - -1 \cdot \frac{z}{y} \]

      3. associate--r+ N/A

         \[\leadsto \color{blue}{1 - \left(\frac{x}{y} + -1 \cdot \frac{z}{y}\right)} \]

      4. mul-1-neg N/A

         \[\leadsto 1 - \left(\frac{x}{y} + \color{blue}{\left(\mathsf{neg}\left(\frac{z}{y}\right)\right)}\right) \]

      5. sub-neg N/A

         \[\leadsto 1 - \color{blue}{\left(\frac{x}{y} - \frac{z}{y}\right)} \]

      6. div-sub N/A

         \[\leadsto 1 - \color{blue}{\frac{x - z}{y}} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{1 - \frac{x - z}{y}} \]

      8. lower-/.f64 N/A

         \[\leadsto 1 - \color{blue}{\frac{x - z}{y}} \]

      9. lower--.f64 98.7

         \[\leadsto 1 - \frac{\color{blue}{x - z}}{y} \]

   5. Applied rewrites 98.7%

      \[\leadsto \color{blue}{1 - \frac{x - z}{y}} \]

   6. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{\frac{z}{y}} \]
   7. Step-by-step derivation

   8. Applied rewrites 97.4%

      \[\leadsto \frac{z}{y} + \color{blue}{1} \]

   if 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x - y}{y}} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{x}{y} - \frac{y}{y}\right)} \]

      2. sub-neg N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{x}{y} + \left(\mathsf{neg}\left(\frac{y}{y}\right)\right)\right)} \]

      3. *-inverses N/A

         \[\leadsto -1 \cdot \left(\frac{x}{y} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto -1 \cdot \left(\frac{x}{y} + \color{blue}{-1}\right) \]

      5. +-commutative N/A

         \[\leadsto -1 \cdot \color{blue}{\left(-1 + \frac{x}{y}\right)} \]

      6. distribute-lft-in N/A

         \[\leadsto \color{blue}{-1 \cdot -1 + -1 \cdot \frac{x}{y}} \]

      7. metadata-eval N/A

         \[\leadsto \color{blue}{1} + -1 \cdot \frac{x}{y} \]

      8. mul-1-neg N/A

         \[\leadsto 1 + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)} \]

      9. unsub-neg N/A

         \[\leadsto \color{blue}{1 - \frac{x}{y}} \]

      10. lower--.f64 N/A

          \[\leadsto \color{blue}{1 - \frac{x}{y}} \]

      11. lower-/.f64 58.3

          \[\leadsto 1 - \color{blue}{\frac{x}{y}} \]

   5. Applied rewrites 58.3%

      \[\leadsto \color{blue}{1 - \frac{x}{y}} \]

   6. Taylor expanded in x around inf

      \[\leadsto -1 \cdot \color{blue}{\frac{x}{y}} \]
   7. Step-by-step derivation

   8. Applied rewrites 58.2%

      \[\leadsto \frac{-x}{\color{blue}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 68.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- z y))))
   (if (<= t_0 2e-15) (/ x z) (if (<= t_0 2.0) 1.0 (/ (- x) y)))))

C

double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double tmp;
	if (t_0 <= 2e-15) {
		tmp = x / z;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = -x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - y) / (z - y)
    if (t_0 <= 2d-15) then
        tmp = x / z
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = -x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double tmp;
	if (t_0 <= 2e-15) {
		tmp = x / z;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = -x / y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x - y) / (z - y)
	tmp = 0
	if t_0 <= 2e-15:
		tmp = x / z
	elif t_0 <= 2.0:
		tmp = 1.0
	else:
		tmp = -x / y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_0 <= 2e-15)
		tmp = Float64(x / z);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(-x) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_0 <= 2e-15)
		tmp = x / z;
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = -x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-15], N[(x / z), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, N[((-x) / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 2.0000000000000002e-15

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 62.6

         \[\leadsto \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\frac{x}{z}} \]

   if 2.0000000000000002e-15 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 94.7%

      \[\leadsto \color{blue}{1} \]

   if 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x - y}{y}} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{x}{y} - \frac{y}{y}\right)} \]

      2. sub-neg N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{x}{y} + \left(\mathsf{neg}\left(\frac{y}{y}\right)\right)\right)} \]

      3. *-inverses N/A

         \[\leadsto -1 \cdot \left(\frac{x}{y} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto -1 \cdot \left(\frac{x}{y} + \color{blue}{-1}\right) \]

      5. +-commutative N/A

         \[\leadsto -1 \cdot \color{blue}{\left(-1 + \frac{x}{y}\right)} \]

      6. distribute-lft-in N/A

         \[\leadsto \color{blue}{-1 \cdot -1 + -1 \cdot \frac{x}{y}} \]

      7. metadata-eval N/A

         \[\leadsto \color{blue}{1} + -1 \cdot \frac{x}{y} \]

      8. mul-1-neg N/A

         \[\leadsto 1 + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)} \]

      9. unsub-neg N/A

         \[\leadsto \color{blue}{1 - \frac{x}{y}} \]

      10. lower--.f64 N/A

          \[\leadsto \color{blue}{1 - \frac{x}{y}} \]

      11. lower-/.f64 58.3

          \[\leadsto 1 - \color{blue}{\frac{x}{y}} \]

   5. Applied rewrites 58.3%

      \[\leadsto \color{blue}{1 - \frac{x}{y}} \]

   6. Taylor expanded in x around inf

      \[\leadsto -1 \cdot \color{blue}{\frac{x}{y}} \]
   7. Step-by-step derivation

   8. Applied rewrites 58.2%

      \[\leadsto \frac{-x}{\color{blue}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 68.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z - y}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-15} \lor \neg \left(t\_0 \leq 5 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- z y))))
   (if (or (<= t_0 2e-15) (not (<= t_0 5e+15))) (/ x z) 1.0)))

C

double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double tmp;
	if ((t_0 <= 2e-15) || !(t_0 <= 5e+15)) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - y) / (z - y)
    if ((t_0 <= 2d-15) .or. (.not. (t_0 <= 5d+15))) then
        tmp = x / z
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x - y) / (z - y);
	double tmp;
	if ((t_0 <= 2e-15) || !(t_0 <= 5e+15)) {
		tmp = x / z;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x - y) / (z - y)
	tmp = 0
	if (t_0 <= 2e-15) or not (t_0 <= 5e+15):
		tmp = x / z
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if ((t_0 <= 2e-15) || !(t_0 <= 5e+15))
		tmp = Float64(x / z);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x - y) / (z - y);
	tmp = 0.0;
	if ((t_0 <= 2e-15) || ~((t_0 <= 5e+15)))
		tmp = x / z;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 2e-15], N[Not[LessEqual[t$95$0, 5e+15]], $MachinePrecision]], N[(x / z), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 2.0000000000000002e-15 or 5e15 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 60.1

         \[\leadsto \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 60.1%

      \[\leadsto \color{blue}{\frac{x}{z}} \]

   if 2.0000000000000002e-15 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5e15

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 93.7%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 2 \cdot 10^{-15} \lor \neg \left(\frac{x - y}{z - y} \leq 5 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8500000 \lor \neg \left(y \leq 9 \cdot 10^{-52}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -8500000.0) (not (<= y 9e-52))) (- 1.0 (/ x y)) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -8500000.0) || !(y <= 9e-52)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-8500000.0d0)) .or. (.not. (y <= 9d-52))) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -8500000.0) || !(y <= 9e-52)) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -8500000.0) or not (y <= 9e-52):
		tmp = 1.0 - (x / y)
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -8500000.0) || !(y <= 9e-52))
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -8500000.0) || ~((y <= 9e-52)))
		tmp = 1.0 - (x / y);
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -8500000.0], N[Not[LessEqual[y, 9e-52]], $MachinePrecision]], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.5e6 or 9.0000000000000001e-52 < y

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x - y}{y}} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{x}{y} - \frac{y}{y}\right)} \]

      2. sub-neg N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{x}{y} + \left(\mathsf{neg}\left(\frac{y}{y}\right)\right)\right)} \]

      3. *-inverses N/A

         \[\leadsto -1 \cdot \left(\frac{x}{y} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto -1 \cdot \left(\frac{x}{y} + \color{blue}{-1}\right) \]

      5. +-commutative N/A

         \[\leadsto -1 \cdot \color{blue}{\left(-1 + \frac{x}{y}\right)} \]

      6. distribute-lft-in N/A

         \[\leadsto \color{blue}{-1 \cdot -1 + -1 \cdot \frac{x}{y}} \]

      7. metadata-eval N/A

         \[\leadsto \color{blue}{1} + -1 \cdot \frac{x}{y} \]

      8. mul-1-neg N/A

         \[\leadsto 1 + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{y}\right)\right)} \]

      9. unsub-neg N/A

         \[\leadsto \color{blue}{1 - \frac{x}{y}} \]

      10. lower--.f64 N/A

          \[\leadsto \color{blue}{1 - \frac{x}{y}} \]

      11. lower-/.f64 75.3

          \[\leadsto 1 - \color{blue}{\frac{x}{y}} \]

   5. Applied rewrites 75.3%

      \[\leadsto \color{blue}{1 - \frac{x}{y}} \]

   if -8.5e6 < y < 9.0000000000000001e-52

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{z}} \]
   4. Step-by-step derivation

      1. lower-/.f64 73.9

         \[\leadsto \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 73.9%

      \[\leadsto \color{blue}{\frac{x}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8500000 \lor \neg \left(y \leq 9 \cdot 10^{-52}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.0% accurate, 18.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 1.0)

C

double code(double x, double y, double z) {
	return 1.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0
end function

Java

public static double code(double x, double y, double z) {
	return 1.0;
}

Python

def code(x, y, z):
	return 1.0

Julia

function code(x, y, z)
	return 1.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0;
end

Wolfram

code[x_, y_, z_] := 1.0

Derivation

1. Initial program 100.0%

   \[\frac{x - y}{z - y} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 33.7%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Developer Target 1: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{x}{z - y} - \frac{y}{z - y} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (/ x (- z y)) (/ y (- z y))))

C

double code(double x, double y, double z) {
	return (x / (z - y)) - (y / (z - y));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x / (z - y)) - (y / (z - y))
end function

Java

public static double code(double x, double y, double z) {
	return (x / (z - y)) - (y / (z - y));
}

Python

def code(x, y, z):
	return (x / (z - y)) - (y / (z - y))

Julia

function code(x, y, z)
	return Float64(Float64(x / Float64(z - y)) - Float64(y / Float64(z - y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x / (z - y)) - (y / (z - y));
end

Wolfram

code[x_, y_, z_] := N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] - N[(y / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024313 
(FPCore (x y z)
  :name "Graphics.Rasterific.Shading:$sgradientColorAt from Rasterific-0.6.1"
  :precision binary64

  :alt
  (! :herbie-platform default (- (/ x (- z y)) (/ y (- z y))))

  (/ (- x y) (- z y)))